How to reach this step?

Question

I failed to understand how the book reached the step with the red border. So any help?

I know that maybe we need to multiply both sides with $\ln$ but where did the $e$ go?

Thanks in advance.

Answer 1

You don’t “multiply” logs to both sides, as that simply makes no sense. You apply them to both sides, or you “take the log of both sides.”

$$8 = 7.1e^{10k}$$

$$\frac{8}{7.1} = e^{10k}$$

Here, you apply the definition of the natural log:

$$e^a = b \iff \ln b = a$$

which yields

$$\frac{8}{7.1} = e^{10k} \iff 10k = \ln\left(\frac{8}{7.1}\right)$$

$$k = \frac{\ln\left(\frac{8}{7.1}\right)}{10}$$